The associated Legendre differential equation is a generalization of the Legendre
differential equation given by

(1)

which can be written

(2)

(Abramowitz and Stegun 1972; Zwillinger 1997, p. 124). The solutions to this equation are called the associated
Legendre polynomials (if is an integer), or associated Legendre functions of the first
kind (if is not an integer). The complete solution is

(3)

where is a Legendre
function of the second kind .

The associated Legendre differential equation is often written in a form obtained by setting . Plugging the identities

into (◇) then gives

(8)

(9)

See also Associated Legendre Polynomial ,

Legendre Differential
Equation ,

Legendre Function of
the Second Kind
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References Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, p. 332, 1972. Moon, P. and Spencer, D. E.
Field
Theory for Engineers. New York: Van Nostrand, 1961. Zwillinger,
D. Handbook
of Differential Equations, 3rd ed. Boston, MA: Academic Press, 1997.
Cite this as:
Weisstein, Eric W. "Associated Legendre Differential Equation." From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/AssociatedLegendreDifferentialEquation.html

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